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Properties

Label 8028.2.a.o
Level $8028$
Weight $2$
Character orbit 8028.a
Self dual yes
Analytic conductor $64.104$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8028,2,Mod(1,8028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8028.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.1039027427\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 51x^{14} + 994x^{12} - 9292x^{10} + 42929x^{8} - 90401x^{6} + 69732x^{4} - 16304x^{2} + 448 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} - \beta_{4} q^{7} + \beta_{15} q^{11} + ( - \beta_{7} - 1) q^{13} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{17}+ \cdots + ( - \beta_{7} + 3 \beta_{5} + \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{7} - 12 q^{13} - 18 q^{19} + 22 q^{25} + 6 q^{31} - 34 q^{37} - 56 q^{43} + 2 q^{49} + 2 q^{55} - 4 q^{61} - 46 q^{67} - 12 q^{73} - 50 q^{79} - 42 q^{85} - 8 q^{91} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 51x^{14} + 994x^{12} - 9292x^{10} + 42929x^{8} - 90401x^{6} + 69732x^{4} - 16304x^{2} + 448 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7381 \nu^{14} + 466475 \nu^{12} - 10979054 \nu^{10} + 116816612 \nu^{8} - 512132853 \nu^{6} + \cdots - 1544935376 ) / 272827008 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 144155 \nu^{14} - 7306237 \nu^{12} + 140875162 \nu^{10} - 1287276940 \nu^{8} + 5616708891 \nu^{6} + \cdots - 725980784 ) / 545654016 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 155459 \nu^{14} - 7735717 \nu^{12} + 145511674 \nu^{10} - 1290158668 \nu^{8} + 5484957315 \nu^{6} + \cdots - 558158000 ) / 545654016 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 149807 \nu^{14} + 7520977 \nu^{12} - 143193418 \nu^{10} + 1288717804 \nu^{8} + \cdots - 994892656 ) / 272827008 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 345085 \nu^{15} - 17991419 \nu^{13} + 362790566 \nu^{11} - 3587202932 \nu^{9} + \cdots - 16097195728 \nu ) / 1091308032 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 201593 \nu^{14} + 10097935 \nu^{12} - 191218558 \nu^{10} + 1701759364 \nu^{8} + \cdots - 41346736 ) / 272827008 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32393 \nu^{14} - 1632985 \nu^{12} + 31247284 \nu^{10} - 282983608 \nu^{8} + 1229459649 \nu^{6} + \cdots - 107585672 ) / 22735584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43925 \nu^{15} + 2202364 \nu^{13} - 41801497 \nu^{11} + 373809646 \nu^{9} + \cdots - 163633300 \nu ) / 68206752 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 432131 \nu^{14} + 21940213 \nu^{12} - 424441450 \nu^{10} + 3913592620 \nu^{8} + \cdots + 1311185264 ) / 272827008 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 922363 \nu^{15} - 46084877 \nu^{13} + 868529450 \nu^{11} - 7643501036 \nu^{9} + \cdots + 20105366864 \nu ) / 1091308032 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 416105 \nu^{15} + 21162379 \nu^{13} - 410892994 \nu^{11} + 3820495612 \nu^{9} + \cdots + 4321984736 \nu ) / 272827008 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 691203 \nu^{15} + 35250757 \nu^{13} - 686852122 \nu^{11} + 6413857804 \nu^{9} + \cdots + 7477748272 \nu ) / 363769344 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2965141 \nu^{15} - 150940307 \nu^{13} + 2933458502 \nu^{11} - 27297527348 \nu^{9} + \cdots - 46894049680 \nu ) / 1091308032 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1986779 \nu^{15} + 101063485 \nu^{13} - 1961608282 \nu^{11} + 18205994764 \nu^{9} + \cdots + 22392930416 \nu ) / 545654016 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - 2\beta_{13} + \beta_{12} - 2\beta_{11} - 2\beta_{9} - 3\beta_{6} + 15\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{10} - 7\beta_{8} - 4\beta_{7} + 13\beta_{5} + 19\beta_{4} + 21\beta_{3} - \beta_{2} + 65 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 23\beta_{15} + 16\beta_{14} - 49\beta_{13} + 21\beta_{12} - 41\beta_{11} - 49\beta_{9} - 56\beta_{6} + 230\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -49\beta_{10} - 167\beta_{8} - 107\beta_{7} + 185\beta_{5} + 293\beta_{4} + 375\beta_{3} - 11\beta_{2} + 840 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 388 \beta_{15} + 221 \beta_{14} - 905 \beta_{13} + 396 \beta_{12} - 739 \beta_{11} - 953 \beta_{9} + \cdots + 3588 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 905 \beta_{10} - 3106 \beta_{8} - 2107 \beta_{7} + 2715 \beta_{5} + 4329 \beta_{4} + 6401 \beta_{3} + \cdots + 11883 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5908 \beta_{15} + 2802 \beta_{14} - 15338 \beta_{13} + 6998 \beta_{12} - 12750 \beta_{11} + \cdots + 56355 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 15338 \beta_{10} - 53392 \beta_{8} - 37138 \beta_{7} + 40169 \beta_{5} + 63497 \beta_{4} + \cdots + 176680 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 85683 \beta_{15} + 32291 \beta_{14} - 250782 \beta_{13} + 118555 \beta_{12} - 215770 \beta_{11} + \cdots + 888175 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 250782 \beta_{10} - 888825 \beta_{8} - 623208 \beta_{7} + 594701 \beta_{5} + 934139 \beta_{4} + \cdots + 2701667 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1206347 \beta_{15} + 317522 \beta_{14} - 4031491 \beta_{13} + 1957745 \beta_{12} - 3615535 \beta_{11} + \cdots + 14032940 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 4031491 \beta_{10} - 14582723 \beta_{8} - 10214801 \beta_{7} + 8794489 \beta_{5} + 13819141 \beta_{4} + \cdots + 41993950 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 16604778 \beta_{15} + 2022055 \beta_{14} - 64288869 \beta_{13} + 31855514 \beta_{12} + \cdots + 222217888 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.03604
−3.80900
−3.02632
−2.52551
−1.85059
−0.927169
−0.590898
−0.177675
0.177675
0.590898
0.927169
1.85059
2.52551
3.02632
3.80900
4.03604
0 0 0 −4.03604 0 −0.887087 0 0 0
1.2 0 0 0 −3.80900 0 −5.05083 0 0 0
1.3 0 0 0 −3.02632 0 −3.48426 0 0 0
1.4 0 0 0 −2.52551 0 3.90049 0 0 0
1.5 0 0 0 −1.85059 0 0.427128 0 0 0
1.6 0 0 0 −0.927169 0 1.58806 0 0 0
1.7 0 0 0 −0.590898 0 −0.254514 0 0 0
1.8 0 0 0 −0.177675 0 0.761016 0 0 0
1.9 0 0 0 0.177675 0 0.761016 0 0 0
1.10 0 0 0 0.590898 0 −0.254514 0 0 0
1.11 0 0 0 0.927169 0 1.58806 0 0 0
1.12 0 0 0 1.85059 0 0.427128 0 0 0
1.13 0 0 0 2.52551 0 3.90049 0 0 0
1.14 0 0 0 3.02632 0 −3.48426 0 0 0
1.15 0 0 0 3.80900 0 −5.05083 0 0 0
1.16 0 0 0 4.03604 0 −0.887087 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(223\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8028.2.a.o ✓ 16
3.b odd 2 1 inner 8028.2.a.o ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8028.2.a.o ✓ 16 1.a even 1 1 trivial
8028.2.a.o ✓ 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8028))\):

\( T_{5}^{16} - 51T_{5}^{14} + 994T_{5}^{12} - 9292T_{5}^{10} + 42929T_{5}^{8} - 90401T_{5}^{6} + 69732T_{5}^{4} - 16304T_{5}^{2} + 448 \) Copy content Toggle raw display
\( T_{7}^{8} + 3T_{7}^{7} - 24T_{7}^{6} - 45T_{7}^{5} + 130T_{7}^{4} + 28T_{7}^{3} - 94T_{7}^{2} + 8T_{7} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 51 T^{14} + \cdots + 448 \) Copy content Toggle raw display
$7$ \( (T^{8} + 3 T^{7} - 24 T^{6} + \cdots + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 105 T^{14} + \cdots + 16128 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} + \cdots - 144)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 176 T^{14} + \cdots + 40656700 \) Copy content Toggle raw display
$19$ \( (T^{8} + 9 T^{7} + \cdots - 70146)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 348509952 \) Copy content Toggle raw display
$29$ \( T^{16} - 332 T^{14} + \cdots + 69722352 \) Copy content Toggle raw display
$31$ \( (T^{8} - 3 T^{7} + \cdots + 672278)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 17 T^{7} + \cdots + 10363)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 359 T^{14} + \cdots + 681408 \) Copy content Toggle raw display
$43$ \( (T^{8} + 28 T^{7} + \cdots + 2645118)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5400958349808 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3715536516412 \) Copy content Toggle raw display
$59$ \( T^{16} - 204 T^{14} + \cdots + 10902528 \) Copy content Toggle raw display
$61$ \( (T^{8} + 2 T^{7} + \cdots + 987840)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 23 T^{7} + \cdots + 20480)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1630913937408 \) Copy content Toggle raw display
$73$ \( (T^{8} + 6 T^{7} + \cdots + 12500)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 25 T^{7} + \cdots + 128512)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 13\!\cdots\!07 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 13347961679808 \) Copy content Toggle raw display
$97$ \( (T^{8} + T^{7} + \cdots + 50400)^{2} \) Copy content Toggle raw display
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